| L(s) = 1 | − 2-s + 4-s − 5-s + 5.12·7-s − 8-s + 10-s + 4·11-s + 4.56·13-s − 5.12·14-s + 16-s − 17-s − 2.56·19-s − 20-s − 4·22-s + 5.12·23-s + 25-s − 4.56·26-s + 5.12·28-s + 5.68·29-s − 6.56·31-s − 32-s + 34-s − 5.12·35-s − 7.12·37-s + 2.56·38-s + 40-s − 4.24·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.93·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 1.26·13-s − 1.36·14-s + 0.250·16-s − 0.242·17-s − 0.587·19-s − 0.223·20-s − 0.852·22-s + 1.06·23-s + 0.200·25-s − 0.894·26-s + 0.968·28-s + 1.05·29-s − 1.17·31-s − 0.176·32-s + 0.171·34-s − 0.865·35-s − 1.17·37-s + 0.415·38-s + 0.158·40-s − 0.663·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.626770750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.626770750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 - 0.561T + 53T^{2} \) |
| 59 | \( 1 - 0.315T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 + 9.12T + 67T^{2} \) |
| 71 | \( 1 - 4.31T + 71T^{2} \) |
| 73 | \( 1 + 6.80T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 - 9.68T + 89T^{2} \) |
| 97 | \( 1 + 1.68T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.075203173756692716550858109519, −8.650294268130628912579324518705, −8.116785050385444878996544820072, −7.16390858275733071543870830653, −6.45273084246191130223227382117, −5.28559653649515123166085993400, −4.40107237274766377165767595559, −3.47608696198409190844338719194, −1.87466930906375217031317439819, −1.13853629183674090390091757946,
1.13853629183674090390091757946, 1.87466930906375217031317439819, 3.47608696198409190844338719194, 4.40107237274766377165767595559, 5.28559653649515123166085993400, 6.45273084246191130223227382117, 7.16390858275733071543870830653, 8.116785050385444878996544820072, 8.650294268130628912579324518705, 9.075203173756692716550858109519
